Cheat Sheet: Properties of Probability Distributions

Here is a probability distribution cheat sheet that I like to keep around for reference. This focuses on the “big picture” properties of some well known PDFs. The goal is to collect some properties that can help me decide when it’s appropriate to use a particular distribution.

has a Student’s t-distribution where Z is standard normal, and V is Chi-Squared with v degrees of freedom.

Is the fat tailed version of the normal distribution.

Useful for characterizing RVs that contain a ratio where small values in the denominator can cause significant weight in the tails.

Arises from looking at a random variable that takes the ratio of the sample mean over the sample standard deviation. In the mathematics the dependency on the actual standard deviation drops away. This is fortunate since during an experiment the actual standard deviation is rarely known.